Foundations for Functions

Transcription

1 Activity: TEKS: Overview: Materials: Grouping: Time: Crime Scene Investigation (A.2) Foundations for functions. The student uses the properties and attributes of functions. The student is expected to: (D) collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations. Students collect data and use scatterplots, correlations, and line of best fit to model the relationship between forearm length and height. Then they analyze the results and predict heights to help solve a crime. CSI Data Collection Sheet CSI Crime Scene Investigation CSI Questions Graphing calculator Yardstick Copy of a teacher s forearm Pairs 30 minutes Lesson: Procedures 1. Provide each student a copy of CSI-Crime Scene Investigation. Read the crime scenario and discuss the problem. 2. Hand out CSI Data Collection Sheet and CSI Questions. Have the pairs discuss how they will collect their data. Each student is responsible for his/her individual scatterplots and solutions to the questions. 3. Students are to determine the independent and dependent variables, the domain, range and scale before beginning the scatterplot. Notes Encourage pairs to discuss the activity as they create their own graphs and answer the questions. The teacher can discuss these properties with the whole class or check student s graphs as they walk the room and monitor the activity. Independent variable forearm Crime Scene Investigations Page 1

2 Procedures 4. Have students answer Questions 2, 3, and 4 on CSI Questions. Notes length; dependent variable height; etc. Depending on students familiarity with the technology, teachers should help students compute their regression equations and r- values. To show the r-value when the regression function is calculated students should select DiagnosticOn from the CATALOG menu before computing the regression function. 5. Have students answer Question 4 on CSI Questions. As a fun challenge, once the students have made their height predictions, have them try to figure out the identity of the owner of the forearm. (Possibly provide some additional clues.) The teacher should prepare a photocopy of a forearm prior to class. Select someone that most likely will fall outside the data found in class. (i.e., a fellow teacher that is very tall or short.) This will allow the students to extrapolate. Also have students use their functions to interpolate. Use some of the students in your class to provide data. Discuss the accuracy of their models and the relationship to their r-values. In summary, discuss the process of using mathematics to solve problems. 1. Ask a question 2. Collect and organize data 3. Make and interpret scatterplots 4. Interpret the correlations 5. Find a model 6. Make a prediction that helps answer your question from step 1. Crime Scene Investigations Page 2

3 Procedures 6. Further questions: What would increase your confidence in your prediction? Suppose you measure another 8 people's forearms and heights and find a regression line to that data. Would the new line be the same as before? Which of these lines should you use to predict height from forearm length? How reliable is our equation for predicting any person's height given we know their forearm length? Notes Collecting more data More accurate measurements A higher r-value These questions of reliability are very important in applying the ideas of this activity to data from any real life experiment. Crime Scene Investigations Page 3

4 CSI - Crime Scene Investigation Consider the following. At approximately 6:45 a.m., Tuesday morning, your principal saw something strange as he opened the backdoor to the high school. As he entered the hallway, he immediately discovered the broken glass from a classroom door. It was a 9 th grade math classroom. The computers were missing, the desks were overturned, and the prized school banner was torn from the wall. The perpetrators were long gone, but they had left something behind. Next to the desk, where the teacher s computer once sat, was the imprint of a forearm on the board. When the police arrived, they immediately began to gather forensic evidence. The principal, knowing your love of CSI and Numb3rs, asks you to help gather data to help identify the bandits. Bones of the arm can reveal interesting facts about an individual. But can they reveal a person's height? Forensic anthropologists team up with law enforcers to help solve crimes. Let's combine math with forensics to see how. Collect data for 8 people. Person Forearm Length (inches) Height (inches) Crime Scene Investigations Page 4

5 CSI Questions 1. From the CSI Data Collection Sheet, describe any relationships you see between the variables forearm length and height. Making a scatter plot can provide a useful visual picture of the relationship between the two variables. 2. On your calculator enter your forearm length data into L1 and corresponding heights into L2 and make a scatterplot. (Make sure you choose a good window for the data.) A positive association is indicated on a scatterplot by an upward trend (positive slope), where larger x-values correspond to larger y-values and smaller x-values correspond to smaller y-values. A negative association would be indicated by the opposite effect (negative slope) where the higher x-values would have lower y-values. Or, there might not be any notable linear association. 3. What kind of association does your data have? In 1896, Karl Pearson gave the formula for calculating the linear correlation coefficient known as r. He argued that it was the best indicator of linear relationships. It derives its name from linear, meaning straight line, and co-relation meaning to "go together." The drudgery of computing them by hand is quite difficult. However, today s calculators can easily compute them. It is often referred to as the Pearson Product Moment Correlation. 4. Use your calculator to compute r for your data. r = We generally categorize the strength of correlation as follows: Strong r > 0.8 Moderate: 0.5< r <0.8 Weak: r < 0. Crime Scene Investigations Page 5

6 If variables are strongly correlated, we can use one to predict the other. A gross example from forensic science is using the size and larva stage of maggots to predict time of death of a body. Linear regression is the method used to create these mathematical prediction models. Given X, we can predict Y. 5. If your correlation is strong enough, record the function for your regression line. Y = 6. Ask your teacher for a copy of the police imprint of our assailant s arm. Predict the height of our assailant using your model. (Show your work below.) 7. Your prediction is In real life, mathematics always begins with a question. What do you want to know? This is followed by data collection. Then scatterplots are drawn to give the big picture. If the relationship looks linear, the correlation coefficient is calculated. If the r value is reasonable, a linear function is found that is used to predict what has not been observed, in our case, the height of the assailant. Crime Scene Investigations Page 6

7 CSI Data Collection Sheet Crime Scene Investigations Page 7